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HB31
.M415
no. S'U
f economics
1991
THE EXISTENCE OF SUBGAME- PERFECT EQUILIBRIUM
IN GAMES WITH SIMULTANEOUS MOVES
Christopher Harris
Number 570
December 1990
massachusetts
institute of
technology
-
50 memorial drive
Cambridge, mass. 02139
THE EXISTENCE OF SUBGAME- PERFECT EQUILIBRIUM
IN GAMES WITH SIMULTANEOUS MOVES
Christopher Harris
Number 570 December 1990
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The Existence of Subgame— Perfect Equilibrium
in Games with Simultaneous Moves:
a Case for Extensive— Form Correlation
Christopher Harris
Nuffield College, Oxford
December 1990^
Abstract
The starting point of this paper is a simple, regular, dynamic game in which
subgame-perfect equilibrium fails to exist. Examination of this example shows that
existence would be restored if players were allowed to observe public signals. The main
result of the paper shows that allowing the observation of public signals yields existence,
not only in the example, but also in an extremely large class of dynamic games.
I should like to thank Drew Fudenberg, Daniel Maldoom, Andreu Mas— Colell, Eric
Maskin, Jean— Francois Mertens, Meg Meyer, Jean Tirole, John Vickers, and
participants at seminars and workshops at Oxford, Stonybrook and Harvard for
helpful comments on the earlier paper from which the present paper is drawn.
The present paper is drawn from an earlier paper entitled "The Existence of
Subgame— Perfect Equilibrium with and without Markov Strategies: a Case for
Extensive Form Correlation".
1 Introduction
Consider the following example of a dynamic game. Firms set out with exogenously
specified capacities (which are known to all). In period one they simultaneously choose
investment levels (possibly on a random basis), and are then informed of one another's
choices. The result is a change in capacities. In period two firms simultaneously choose
production levels (within their capacity constraints), and are then informed of one
another's choices. The result is a change in inventories. In period three firms
simultaneously choose prices. The vector of prices chosen affects the vector of demands for
their products, but so do certain exogenous random factors. Firms are informed of one
another's chosen prices and of the final realised demands. (The demands bring about a
second change in inventories.) The three period cycle of choices the begins afresh. And so
on.
This game is an example of a game of the following general type. Time is discrete.
There is a finite number of active players. There is also a passive player, Nature. In any
given period: all players (both active and passive) know the outcomes of all previous
periods; the set of actions available to any active player is compact, and depends
continuously on the outcomes of the previous periods; the distribution of Nature's action
(which is given exogenously) depends continuously on the outcomes of the previous periods;
the players (active and passive) choose their actions simultaneously; and the outcome of
the period is simply the vector of actions chosen. The outcome of the game as a whole is
the (possibly infinite) sequence of outcomes of all periods, and players' payoffs are bounded
and depend continuously on the outcome of the game.
This is as regular a class of dynamic games as one could ask for. A counter-example
shows, however, that games in this class need not have a subgame— perfect equilibrium. It
is therefore necessary to extend the equilibrium concept in such a way that existence is
restored. l
The problem is reminiscent of that of the non-existence of Nash equilibrium in pure
What is the most natural extension of the equilibrium concept? One clue as to the
answer to this question is provided by the following considerations. First, if we simplify
the class of games under consideration by requiring that players' action sets are always
finite, then subgame— perfect equilibrium always exists. Hence the non-existence problem
appears to relate specifically to the fact that we allow a continuum of actions. Secondly,
suppose instead that we simplify the class of games under consideration by assuming that
players' action sets are independent of the outcomes of previous periods. Then a natural
way of approximating a game is to consider subsets of players' action sets that consist of
large but finite numbers of closely spaced actions.2 Moreover, if one takes a sequence of
increasingly fine approximations, and a subgame— perfect equilibrium of each of the
approximations, then it is natural to expect that any limit point of the sequence of
equilibrium paths so obtained will be an equilibrium path of the original game.3 One can,
however, find examples in which the limit point involves a special form of extensive— form
correlation: at the outset of each period agents observe a public signal.4
strategies. But there is one important difference: if agents' action sets are always finite,
then subgame— perfect equilibrium does exist.
2 Hell wig and Leininger (1986) were the first to introduce such an approximation, in
the context of finite— horizon games of perfect information. They proved an
upper— semicontinuity result: they showed that any limit point of equilibrium paths of the
finite approximations is an equilibrium path of the original game. Their result can,
however, be understood in terms of the results of Harris (1985a). Indeed, in that paper it is
shown that the point— to— set mapping from period— t subgames of a game of perfect
information to period— t equilibrium paths is upper semicontinuous. Hence, in order to
obtain their result, one need only introduce a dummy player who chooses n e IN U {qd} at
the outset of the game. If n < od then the remaining players will be restricted to actions
chosen from the n approximation to their action sets. If n = od then they will be free to
choose actions from their original action sets.
3 Convergence of equilibrium paths, as used implicitly in Harris (1985a) and explicitly
in Hellwig and Leininger (1986) and Borgers (1989) seems more relevant than the
convergence of strategies considered by Fudenberg and Levine (1983) and Harris (1985b).
4 Fudenberg and Tirole (1985) consider two such games. They point out that the
obvious discretisations of these games have a unique symmetric subgame— perfect
equilibrium, and that the limiting equilibria obtained as the discretisation becomes
arbitrarily fine involve correlation. Their games do not, however, yield counterexamples to
the existence of subgame— perfect equilibrium. For both games possess asymmetric
equilibria which do not involve correlation.
These considerations suggest that, at the very least, the equilibrium concept should
be extended to allow the observation of public signals. This minimal extension is also
sufficient. For, first, it restores existence. Secondly, with it, the equilibrium
correspondence of a continuous family of games is upper semicontinuous. In particular, any
limit, point of equilibrium paths of finite approximations to a game whose action sets are
continua is an equilibrium path of the game. Thirdly, any limit point of subgame-perfect e-
equilibrium paths of a game is an equilibrium path in the extended sense. 5
The basic structure of the proof of the existence of subgame-perfect equilibrium in
the extended sense is the same as the structure of the proof of the existence of
subgame-perfect pure— strategy equilibrium in games of perfect information given in Harris
(1985a). It breaks down into two main parts, the backwards and the forwards programs.
The backwards program is most easily explained in the context of a game with finite
horizon T. In such a game, it consists in solving recursively for what turn out to be the
equilibrium paths of the game. More precisely, one solves first for the equilibrium paths of
period— T subgames. (The set of equilibrium paths of a period— T subgame is the convex
hull of the set of probability distributions over period— T outcomes.) Then one solves for
the equilibrium paths of period— (T— 1) subgames. (The set of equilibrium paths of a
period— (T— 1) subgame is a set of probability distributions over period— (T— 1) histories, i.e.
a joint distribution over period— (T— 1) and period— T outcomes.) And so on until period 1
is reached, and a set of probability distributions over period— 1 histories is obtained.
There is of course no general guarantee that the paths obtained in the course of
carrying out the backwards program really are equilibrium paths until equilibrium
strategies generating them have been constructed. To construct such strategies is the
purpose of the forwards program. The essential idea is this. Pick any of the period— 1
paths obtained as the final product of the backwards program. Such a path is a probability
distribution over period— 1 histories. Its marginal over period— 1 outcomes can be used to
5 Chakrabarti (1988) claims that subgame-perfect e-equilibria exist. We believe this
claim, but are unable to vouch for the proof.
construct period— 1 behaviour strategies for the players. Its conditional over period— 2
histories yields continuation paths to be followed from period 2 onwards in the event that
players do not deviate from their period— 1 strategies. In order to obtain continuation
paths in the event that some player does deviate from his period— 1 strategy, it is sufficient
to choose from among the period— 2 paths obtained in the course of the backwards program
some path which minimises the continuation payoff of that player. In this way
continuation paths for all period— 2 subgames are obtained. Period— 2 strategies are then
obtained from the marginals of these paths over period— 2 outcomes. And so on until
finally period— T strategies are obtained.
Implementing this proof does, however, involve two significant new complications.
First, for technical reasons, it is necessary to maintain the induction that the point— to— set
mapping from period— t subgames to period— t equilibrium paths is upper semicontinuous.
This follows from a generalisation of the work of Simon and Zame (1990). Secondly,
because players may randomise, it is essential to ensure that all the strategies constructed
are measurable. 6 That this requirement can be met emerges as a natural corollary of the
construction employed: one can always construct a measurable family of conditional
distributions from a measurable family of probability distributions.7
The organisation of the paper is as follows. Section 2 sets out the counterexample
to the existence of subgame— perfect equilibrium. This example involves a three— period
game with two players in each period. In this game, each player's action set is compact
6 Hellwig and Leininger (1987) were the first to draw attention to the desirability of
ensuring that strategies are measurable. Their approach to the existence of
subgame— perfect equilibrium is not, however, necessary in order to obtain measurability.
Indeed, if the assumptions of Harris (1985a) are specialised appropriately (by assuming
that the embedding spaces for players' action sets are compact metric instead of compact
Hausdorff), then it is simple to show that exactly the construction given there can be
carried out in a measurable way. Indeed, in the notation of Harris (1985a; p. 625), all that
is necessary is to ensure that the functions gj can be chosen to be measurable; and that
this is possible follows from the argument given in the first paragraph of the proof of
Lemma 4.1 in the present paper.
7 Had we employed a dynamic— programming approach, the problem of ensuring
measurability of strategies would have been much harder — see below for further discussion.
Indeed, we do not know how to solve the measurable selection problem that arises when
such an approach is employed.
and independent of the outcomes of previous periods, and each player's payoff function is
continuous. Section 3 formulates the basic framework used in this paper for the analysis of
dynamic games. Section 4 shows that when the dynamics are continuous, players' payoff
functions are continuous, and the framework is extended to allow the observation of public
signals, subgame— perfect equilibrium exists. This is the main result of the paper. Section
4.1 provides a detailed overview of the proof. Section 4.2 develops the analysis necessary
for the backwards program. Section 4.3 develops the analysis necessary for the forwards
program. Section 4.4 exploits the analysis of Sections 4.2 and 4.3 to prove the main result,
showing in particular how to deal with the infinite— horizon case. Section 5 attempts to
develop a perspective on the main result. It is shown there that there are at least three
types of game in which the introduction of public signals is not necessary to obtain the
existence of subgame-perfect equilibrium: games of perfect information, finite-action games,
and zero-sum games. It is also shown that the introduction of public signals accommodates
all the equilibria that one can obtain by taking finite-action approximations to a game, or
by considering e— equilibria of a game. We do not know whether the introduction of public
signals is the minimal extension with these two properties, but it seems plausible to argue
that it is.
2 The Counterexample
A counterexample to the existence of subgame— perfect equilibrium in the class of
dynamic games described in the introduction must have a certain minimum complexity.
First of all, it must have at least two periods. Otherwise the standard existence theorem
for Nash equilibrium would apply. Secondly, there must be at least two players in the
second period. For with only one player in the second period, the correspondence
describing the continuation payoffs available to the player or players in period one would
be convex valued. (The player in period two can randomise over any set of alternatives
between which he is indifferent.) So the existence result of Simon and Zame (1990) would
apply to period one. Thirdly, there must be at least two players in period one. For the
continuation— payoff correspondence for period one will be upper semicontinuous, and a
single player will therefore be able to find an optimum.
The counterexample presented in this section is not this minimal. It has three
periods, with two players in each period. It would be of considerable interest to find an
example with only two periods and two players in each period. For, aside from being
minimal in relation to the class of dynamic games considered in this paper, such an
example would settle the question as to whether equilibrium exists in two— stage games or
not.
The cast of the counter— example, and their choice variables, are as follows. In
period one two punters A and B pick a 6 [0,1] and b € [0,1] respectively. In period
two, two greyhounds C and D each receive an injection of size a + b , which changes
their attitude to the race. They pick c e [0,1] and d 6 [0,1] , which are the times in which
they complete the course. In period three each of two referees E and F must declare a
winner, picking e 6 {C,D} and f e {C,D} respectively. In each period choices are made
simultaneously, and players in later periods observe the actions taken by players in earlier
periods.
Punter A obtains a payoff of 1 — a if greyhound C is declared the winner by
both referees, and — 1 — a otherwise. Similarly, punter B obtains 1 — b if both referees
declare D to be the winner, and — 1 — b otherwise. In other words, A wants C to win,
B wants D to win, and both want a result. They would also like to keep their
contributions to the injection as small as possible. The payoff to greyhound C is 2c if
e = C and 1 — (a + b)(l — c) if e = D . That is, the form of his payoff depends on
whether he or the other greyhound is declared the winner by referee E , but either way he
would prefer to run the race as slowly as possible. Also, he would prefer to be first rather
than second provided that he does not have to run too fast. The payoff to greyhound D is
2d if e = D and 1 — (a + b)(l — d) if e = C : like greyhound C , he is only interested
in the verdict of referee E . Lastly, referee E gets payoff d is he declares C to be the
winner and c if he declares D to be the winner. Referee F 's payoffs are identical.
It will be seen that, in this game, each player's action set is compact and
independent of the actions chosen by earlier players, and that players' payoffs are
continuous.
The game centres around the two greyhounds. Because referee E chooses C when
c < d and D when c > d , the race between them is essentially a game of timing with
the discontinuous payoffs (2c, 1 — (a + b)(l — d)) if c < d , (1 - (a + b)(l - c), 2d) if
c > d , and some convex combination of these two payoffs when c = d . (Note that the
weights in the convex combination can depend on (c,d) .) Standard considerations
therefore yield the following lemma.
Lemma 2.1 Suppose that a + b > 0 . Then both C and D use mixed actions with
distribution function G given by G(x) = 0 for x e [0,|] and G(x) = (2x - l)/(2x - 1
+ (a + b)(l-x)) for xe [$,1] . □
Notice that this mixed action is non-atomic, that its support is [j,l] , and that all
its probability mass concentrates at £ as a + b -» 0+ . Not surprisingly, then, we obtain:
Lemma 2.2 Suppose that a + b = 0 . Then both C and D choose $ with probability
one. d
The remainder of the argument is straightforward. If a + b > 0 then c will equal
d with probability zero. Hence both referees will always agree. Also, each greyhound will
win exactly half the time. The payoffs to A and B are therefore —a and — b. If, on the
other hand, a + b = 0 , then c = d = \ with probability one. Hence each referee is
indifferent as to which greyhound he declares to be the winner. Suppose that E opts for
C with probability p and D with probability 1 — p , and that F opts for C with
probability q and D with probability 1 - q . Then A 's payoff is 2pq — 1 and B 's is
2(l-p)(l-q)-l.
It is easy to see, however, that the game with these payoffs between A and B has
no equilibrium. Indeed, any a > 0 is strictly dominated for A , as is any b > 0 for B .
So the only possibility for an equilibrium is for A to set a = 0 with probability one and
B to set b = 0 with probability one. But if b = 0 then it must be that 2pq — 1 > 0
— otherwise A would raise a from zero. Similarly, 2(1 — p)(l — q) — 1 > 0 . And these
two inequalities are mutually inconsistent.8 This contradiction establishes the counter-
example.
The essence of the counterexample is this. As long as a + b > 0 , both greyhounds
use strictly mixed actions. This behaviour on their part generates a public signal
endogenously within the game. The two referees use this public signal to co-ordinate their
actions. When a + b = 0 , however, this signal suddenly degenerates, and the only way
in which the referees can coordinate their actions is by both choosing C with probability
one or both choosing D with probability one. But if they both choose C then B gets
8 To see this, note that the sum of the payoffs of the two players A and B is 0 if
e = f and —1 if e ^ f. Hence the expected value of this sum is non— positive. If the
expected value is strictly negative, then the payoff of at least one player is also strictly
negative. If the expected value of the sum is zero, then e and f must be perfectly
correlated. In this case the payoff of the player against whom the decision goes is — 1.
—1 . So he would prefer the truly random outcome. Similarly, if both referees choose D
then A will wish to restore the truly random outcome.
In the light of this explanation, it is natural to try to restore existence by allowing
players to observe suitable public signals. In the present example, this would amount to
allowing the two referees to toss a coin to determine the winner in the event of a tie (which
would certainly restore existence). That such an extension yields existence in the general
case will be demonstrated in Section 4.
10
3 The Basic Framework
3.1 The Data
There is a non-empty finite set J<A of active players, indexed by i or j . There
is also a passive player, Nature, whose index is i or j = 0 . The overall set of players is
therefore J= J^i U {0} . Time is discrete, and is indexed by t or se J"= {0,1,2,...} .
For notational reasons, we assume that J n «J= 0 . The players play one of a family of
games, parameterised by x eX . In each active period t 6 STjd = {1,2,...} each player
i e J must choose an action. The set of actions available to her or him depends on which
game is being played, and on the outcomes of previous periods. This situation is modelled
by a point— to— set mapping A..: X — -» Y,. . The vector x — e X — lists the
parameter of the game being played, and the outcomes of any preceding periods, while
A, .(x ) C Y, - is the set of actions available. For reasons of expositional economy, we
take it that A.^x — ) = Y,^ for all x — € X ~~ .9 The set of outcomes possible in
period t is nothing more than the set of profiles of actions that players can take. It is
modelled by the point-to-set mapping A .: X — -♦ Y. , where Y. = x Y.. and
A,(x ) = x A,.(x ) for all x EX . And X is simply the set of all pairs
(x ~ ,y, ) such that x — eX~ and y, e A,(x — ) , i.e. the graph of A, . Finally, the
payoff of any active player i depends on which game x e X is played, and the
outcomes of all periods. It is modelled by a function u-: Xw -» IR . Here X00 is the set of
,0
t-1 / N . Vt-1
vectors x = (x0,Xp...) e X x x Y. such that x — = (xn,x. ,..,x,_.) eX~ for all
t > 1 . (Nature does not have a payoff function.)
We make the following standing assumptions about these data:
(i) X , and all of the sets Y.. , are non-empty complete separable metric spaces;
(ii) for all i € X4 and all t > 1 , A..: X — -♦ J6(Y,-) is measurable;
(iii) for all i e J<A , u- is bounded and measurable.
9 This assumption does in principle involve a slight loss of generality. However, since
we shall in any case fix Nature's strategy below, there seems to be little advantage in
constraining her actions as well.
11
(In this paper, 'measurable' will always mean Borel measurable unless explicitly stated to
the contrary; and •%(•) will always denote the space of non-empty compact subsets of •
endowed with the Hausdorff metric.) These assumptions are designed to ensure that every
player possesses at least one strategy, and that associated with every strategy profile and
every subgame there is a well defined payoff for every active player (see below). Stronger
assumptions are needed to ensure the existence of equilibrium.
3.2 Strategies and extended strategies
In the standard version of our model, the evolution of the game is as follows. At the
beginning of period 1, players are told which game they are playing. That is, they are
informed of x . They then simultaneously choose actions from their action sets for period
1. This completes period 1. At the beginning of period 2, players are told the outcome of
period 1. That is, they are told y, . They then choose simultaneously from their action
sets for period 2. This completes period 2. And so it goes on. It follows that players'
information in period t can be summarised by a vector x — eX~ , and that X — can
be identified with the set of subgames in period t .
Definition 3.1 For each t > 1 and each i e J , a strategy for player i in period t is a
measurable function f,.: X ~~ -> 9Jl^{4-) such that supp[f,.(x _ )] C A,.(x ) for all
t-1 , vt-l
x 6 X
Here 9Ji{XA denotes the set of probability measures over Y,. . (In this paper,
'probability measure' will always mean Borel probability measure; and spaces of
probability measures will be taken to be endowed with the weak topology10 unless explicitly
stated to the contrary.)
10 In the terminology of Parthasarathy (1967), a sequence of probability measures
{A } C &M(Y.) converges in the weak topology to A iff Jx dXn converges to /xdA for
all continuous bounded x- Yf -» IR .
12
The f.- are behaviour strategies. For each t , i and x , f, .(x ) can be
thought of as the randomising device that player i will use to choose among the actions
available to him in period t when x is the previous history of the game. As such, it is
independent of the devices used by the other players in period t , and of all devices used in
all preceding and subsequent periods.
Strategies and strategy profiles certainly exist under our standing assumptions.
Moreover, given any strategy profile and any subgame, the payoffs of active players can be
calculated in the natural way. We may therefore define subgame— perfect equilibrium as
follows.
Definition 3.2 A subgame— perfect equilibrium is a strategy profile <f,- 1 1 > 1, i e Jj&>
such that, for all t > 1, all x — e X — , and all i G X^" , player i cannot improve his
payoff in subgame x by a unilateral change in his strategy.
As the counterexample of Section 2 shows, however, subgame— perfect equilibrium
need not exist in general. We therefore introduce the concept of an extended strategy.
Suppose that {£. |t > 1} is a sequence of signals, drawn independently from [0,1]
according to the uniform distribution. Suppose that, at the beginning of each active period
t > 1 , players are informed not only of the outcome of the preceding period, but also of
£, . Then the information available to them when they make their choices can be
summarised by a vector h — = (x — ,£ ) , where £ = (£-,,..., £t) ■ We refer to the set
H ~~ = X — x [0,1] of such vectors as the set of extended subgames of the game in
period t .
Definition 3.3 For each t > 1 and each i e J , an extended strategy for player i in
period t is a measurable function f,.: H — -> &J((Y..) such that supp[f,.(x — ,£ )]
C Ati(xt_1) for all (xt_1,^) e Ht_1.
13
Once again the f,. are behaviour strategies. So f..(x — ,£ ) can be thought of as
the randomising device that player i will use to choose among the actions available to him
in period t , when x — is the previous history of the game and £ is the vector of public
signals observed up to and including period t . But this time this device is independent,
not only of the devices used by the other players in period t and of all devices used in all
other periods, but also of all the public signals.
Definition 3.4 A subgame— perfect equilibrium in extended strategies is an extended-
strategy profile <f. |t > 1, i e X^> such that, for all t > 1 , all h — eH~ , and all i
6 Jj6 , player i cannot improve his payoff in extended subgame h by a unilateral
change in his strategy.
A subgame— perfect equilibrium in extended strategies allows a limited amount of
coordination: after the outcome of a period has been realised, the players can coordinate on
the continuation equilibrium to be played following that outcome by exploiting the next
public signal. It is therefore a limited kind of extensive— form correlated equilibrium. (The
correlation is limited in that the players observe a common signal, rather than each
privately observing a single component of a vector of signals.)
14
4 The Main Result
We need the following assumptions:
(Al) for all t > 1 and all i e J^i , the mapping A.-: X ~~ -» J£(T..) is continuous;
(A2) for all t > 1 , a strategy f.* (and not an extended strategy) for Nature in period t
is given, and f,*: X — -* 9M (Y,,-.) is continuous;
(A3) for all i G J^i , u- is bounded and continuous.
Taken in conjunction with the standing assumptions made in respect of the basic
framework, they guarantee the existence of a subgame— perfect equilibrium in extended
strategies.
The present section is devoted to a proof of this fact. It begins, in Section 4.1, with
an overview of the proof. The proof involves two main steps. The analysis necessary for
the first of these, namely the backwards program, is given in Section 4.2. The analysis
necessary for the other, namely the forwards program, is given in Section 4.3. Section 4.4
then integrates the backwards and the forwards programs to complete the proof. The
reader may wish to read Section 4.1 and then skip to Section 5.
4.1 Overview
Imagine for the purposes of the present subsection that the game has horizon T .
Then the proof divides into essentially two steps. In the first step, one programs backwards
from stage T , finding what turn out to be the equilibrium paths of the game. In the
second, one programs forwards, constructing strategies that generate these paths and that
constitute an equilibrium.
The basic ideas of the backwards program are as follows. First, for each history
T— 1 T— 1
x e X , find the set of Nash equilibria (including Nash equilibria in mixed
strategies) that can occur in the final stage of the game. Let C™(x ) be the set of
probability distributions over Y™ that result from these Nash equilibria. Because action
T— 1
sets are compact and payoff functions are continuous, C™(x ) is a non-empty compact
15
T— 1
set contained in 9JC{Xrr) . Because actions sets depend continuously on x , the
T— 1
point— to— set mapping C™ from X to ^(^^(Yrp)) is upper semicontinuous.11
•p o
Next, let x be the history of the game prior to period T— 1 . For any v™ -
T— 2
G AT_1 (x ) , the set of continuation paths consistent with equilibrium is given by
<p o
co[CT(x ,yrp_1 )] . (This set consists, in effect, of those probability distributions over
YT that can be obtained by randomising over Nash equilibria in Crp(xrj,_, ) .) For each
selection cT(x , • ) from the correspondence co[Cy(x , • )] , find the set of Nash
equilibria that can occur in period T— 1 when the continuation of the game is specified by
^p 2 T 2
Crp(x , •) . (For a general c,y(x ,•) , this set may be empty. But there will always
<p 2
be at least one c,y,(x ,•) for which it is not. This is essentially the content of the
existence theorem of Simon and Zame (1990).) And for each Nash equilibrium associated
T— 2
with Crp(x , • ) , find the probability distribution over Y™, * Y™ that results when
that equilibrium is played in period T— 1 and the continuation paths are given by
p<_2 "p 2
Crp(x , • ) . Let Crp_-. (x ) be the set of such distributions obtained when both
T— 2
Crp(x , •) , and the Nash equilibrium associated with it, are allowed to vary. Then
T— 2
Cy,_|(x ) is a non-empty compact subset of &J£(Yrr< _i*Yrp) , and the point— to— set
p 2
mapping C™, from X to ^(^^(Y^ixYrp)) is upper semicontinuous. (This is
essentially the content of the generalisation of Simon and Zame's theorem given in
Section 4.2.)
11 The relevance of the assumption that players' action sets in period T depend
continuously on the history of the game prior to period T emerges clearly in the context
of a two— period game. For each n > 1 , let a? be a Nash equilibrium in period 2 of such a
game when the outcome of period 1 is y, , and suppose that (y^a^) -» (y^a^) as n -» <d .
Because player i 's action set is upper semicontinuous in the outcome of period 1, the
action a„j is actually available to him in period 2 when the outcome of period 1 is y-, .
Also, because his action set in period 2 is lower semicontinuous in the outcome of period 1,
any deviation open to him in period 2 when the outcome of period 1 is y, can be
approximated by deviations open to him when the outcome of period 1 is y, . Hence a^.
really is an optimal action.
16
Finally, iterate until period 1 is reached, and a set C.(x ) of probability
T 0 0 0
distributions over X...Y. is obtained for all x eX . (The set co[C,(x )] will turn
out to be the set of equilibrium paths of the game x .) This completes the backwards
program.
Consider now the forwards program. Suppose that, for each x 6 X , we are given
c,(x ) 6 co[C.(x )] . Since any path in co[C,(x )] can be viewed as a randomisation over
paths in C.(x ), we can associate a probability measure d,(x ) over C,(x ) with
c..(x ) . Also, for each d,(x ) we can find a random variable e,(x ,•): [0,1] -» C,(x )
whose distribution over C,(x ) is d,(x ) when [0,1] is given Lebesgue measure. For
each i and each £ e [0,1] , let f,-(x ,£ ) be the marginal of e,(x ,£ ) over Y-. . By
definition of C,(x ), there exists a selection c~(x ,-,£ ) from co[C2(x ,-)] such that:
(i) the f,-(x ,£ ) constitute a Nash equilibrium in period 1 when the continuation paths
are given by c«(x ,-,£ ) ; and (ii) for all y,, c«(x ,y-.,£ ) is the conditional distribution
of e, (x ,£ ) given that y1 is the outcome in period 1.
Similarly, beginning with cJx ,£ ) 6 co[C2(x )] one obtains d„(x ,^ ), eJx ,( ),
12 2 2
f„.(x ,£ ) and cJx ,£ ) . And so on until one has a complete set of strategies f,- for all
1 < t < T and i e J^£ . That these strategies really do produce the paths they are
intended to produce is primarily a technical matter. The essential points are that the
expectation of e,(x - ,£ - ,-) is always c,(x — ,f — ) , and that c, , .(x — ,y,,^ ) is
always the conditional distribution of e,(x ,£ ) given y, . That they constitute an
equilibrium follows from the fact that the f,-(x — ,£ ) constitute a Nash equilibrium.
This ensures that no single— period deviation is profitable. That no more complex deviation
is profitable follows from a standard unravelling argument. This completes the forwards
program.
Overall, the essential ingredients of the theorem are the following: players action
sets are compact; players' payoffs are continuous; Nature's strategy is continuous; players'
action sets in a given period depend continuously on the history of the game prior to that
period; players observe public signals; and players' payoffs are bounded. The role played
17
by the first three of these should be obvious. The role played by the fourth is, roughly, to
ensure that the point— to— set mapping from subgames in period t to equilibrium paths of
those subgames is upper semicontinuous. The role of the fifth is to ensure that the
point— to— set mapping from subgames in period t to equilibrium paths of those subgames is,
in addition, convex valued. And the sixth is needed only for the relatively technical reason
that we do not insist that Nature's behaviour strategies have compact support.
4.2 The backwards program
In this subsection we develop the analysis necessary for the backwards program. To
this end:
(i) let X, Y- for all i e J , and Z be non-empty complete separable metric spaces;
(ii) for all i 6 X4 , let A-: X -> Jfc(Y-) be a continuous point— to— set mapping;
(iii) let f*: X -» 9>Jt (Y*) be a continuous mapping;
(iv) let Y = "jgjYj and let A(x) = Y^ x x ^Aj(x) for all x £ X ;
(v) let C : gr(A) -» Jf(Z) be an upper semicontinuous point— to— set mapping;
(vi) for each i e J^i , let u-: gr(C) -» R be a continuous function.
The similarity of this notation to that of Section 3 is deliberate, and is intended to be
helpful, not confusing.
With reference to the model of Section 3 and the discussion of Section 4: X is the set
of past histories; A. describes the dependence of player i 's action set on the past history;
f* describes Nature's behaviour; C describes the set of possible continuation paths; and
u- is the payoff of active player i . In terms of a somewhat more abstract perspective,
what we have is a continuous family of games parameterised by x e X . The three
essential ingredients of such a family are: the continuity of the action sets; the upper
semicontinuity of the continuation paths; and the continuity of the payoffs in the
parameter, the outcome and the continuation.
For each x 6 X we are interested in the set of equilibrium paths
^C(x) c &Jt(Y * Z) that are obtained when one is free to choose any randomisation over
18
C(x,y) as the continuation path following the outcome y G A(x) . More formally, A
€ Jc(x) iff:
(i) the marginal \i of A over Y is a product measure with supp[/x] C A(x) ;
(ii) the marginal /jg of \i over Y* is frt(*|x);
(iii) A possesses an r.c.p.d. (regular conditional probability distribution) v over Z
such that supp[i/( • | y)] C C(x,y) for all y e A(x) ;
(iv) moreover the marginals \i- of (j, over the Y- constitute a Nash equilibrium when
the continuation path following outcome y is v( • | y) for all y .
(It may be helpful to state more explicitly what is meant by (iv). Suppose we are given
probability measures fj,- over A.(x) for all \£ J and a transition probability v such
that supp[^(- |y)] C C(x,y) for all y 6 A(x) . Let A be the measure over Y«Z
obtained by combining the product of the n- with v . Then the payoff of an active player
i is /u-(x,y,z)dA(y,z) , and the \i- constitute a Nash equilibrium if no such player can
improve his payoff by changing his probability measure.)
Lemma 4.1 #C is an upper semicontinuous map from X into Jf( 3>JC{Y * Z)) .
Proof We begin with some notation. For each i , x , and y 6 A(x) , let p-(x,y)
= min{u.(x,y,z) | z 6 C(x,y)} . Then p-(x,y) is the lowest payoff that can be imposed on
player i in game x following outcome y . Standard considerations show that p. is lower
semicontinuous. Let P:(x,y) = {z|z e C(x,y) and u-(x,y,z) = p-(x,y)} . We shall need a
measurable selection from P. . Following Deimling (1985; Proposition 24.3 and Theorem
24.3), the existence of such a selection will follow if P7 (D) = {(x,y)|D n P-(x,y) j- 0} is
measurable for all closed D C Z . To this end, define p ■ (x,y) = min{u-(x,y,z)|z 6 D
n C(x,y)} . (By definition p . (x,y) = oo if D n C(x,y) = 0 .) Like p. , p • is lower
semicontinuous. Hence P~ (D) = {(x,y)|p.(x,y) = p. (x,y)} is measurable. Let q.: gr(A)
-• Z be the resulting measurable selection.
19
The proof now divides into three parts. For each x e X , let B(x) be the set of
paths of game x that are consistent with Nature's mixed action £*( • jx) . The first part of
the proof shows that this mapping is an upper semicontinuous mapping from X into
J6{9Jl^i * Z)) . The second shows that the graph of #C is closed. Since this graph is
clearly contained in that of B , all that remains is to demonstrate that #C is non— empty
valued. This is done in the third step. The second step accounts for the bulk of the proof.
It is clear that B is non-empty valued, and that its graph is closed. All that we
need therefore show is that it is compact valued. To this end, let x e X be given. Since
B(x) is closed, we need only show that it is tight in order to show that it is compact.
That is, we need only show that, for all e > 0 , there exists a compact K c Y * Z such
that A(K) > 1— e for all A e B(x) . But there certainly exists a compact K C Y* such
(V
that ffl(K) > 1— e , and we may therefore let K be the graph of the restriction of C(x, • )
to K
ieX*
AjW
. This completes the first step.
In order to begin the second step, suppose that (x ,A ) 6 gr^C] , and that (x ,A )
n' ny
n' n'
-» (x,A) . For all i , let r- be player i 's payoff from A , and let it. be his payoff from
A . From the definition of ^C(x ) one can deduce that:
/Xi(yi)xj(yj)dAn(y,z) = [/xi(yi)dAn(y,z)J |/xj(yj)dAn(y>z)
.(5.1)
for all i # j and all continuous y. : Y. -» R and y- : Y.
J Ai i AJ J
IX0(y0)dAn(y,z) = jy0(y0)df0(y0]xn)
...(5.2)
for all continuous Xa '■ Y0 -» K ;
supp(An)cgr(C(xn,.))
.(5.3),
20
where C(x , • ) is regarded as a (possibly empty— valued) correspondence from Y to Z
n
(C(xn,y) is empty iff y t A(xn)) ;
/Xi(yi)ui(x,y,z)dAn(y,z) = ^/^(y^dA^z) ...(5.4)
for all active i and all continuous X\ :Y. -» R ; and
rM > /pi(xn,y\ani)dAn(y)z) ...(5.5)
for all active i and all a . e A.(x ) .
ni p n'
Equation (5.1) follows from the independence of the marginals of A over the Y. .
Equation (5.2) follows from the fact that A is consistent with Nature's strategy.
Relation (5.3) follows from the facts that the marginal \i of A is supported on A(x ) ,
and that its conditional u (• |y) is supported on C(x ,y) for all y 6 A(x ) . Equation
(5.4) holds because the expectation of u-(x ,-,-) conditional on y. is ir ■ a.s. This
follows from the fact that, in equilibrium, the probability mass of player i 's mixed
strategy must be confined to a set of actions, each of which yields him his equilibrium
payoff. (It is important to avoid the word 'support' here. For player i 's continuation
payoff /u-(x ,y,z)di/ (z|y) need not be continuous in y , and so the set of pure strategies
yielding him r. need not be closed.) Inequality (5.5) states that player i 's equilibrium
payoff must be at least what he would get if he were to deviate to a • and if, following
this deviation and no matter what the realised choices of the other players, the worst
conceivable continuation path from his point of view occurred. Equation (5.4) captures the
idea that no redistribution of probability mass among equilibrium actions is worthwhile.
Inequality (5.5) captures the idea that no redistribution of probability mass from
equilibrium to out— of— equilibrium actions is worthwhile.
Now it is easy to see that relations (5.1), (5.2) and (5.3) are preserved in the limit.
Also, since ic ■ = /u-(x ,y,z)dA (y,z) , x . -» ir. . Hence (5.4) is preserved too. Lastly, for
21
any a- e A.(x) , there exist a . e A-(x ) such that a . -» a- , by the lower semi continuity
of A. . Since also w- is lower semicontinuous, (5.5) too is preserved. The proof that #C
has a closed graph will therefore be complete if we can show that the analogues of (5.1)-
(5.5) for x and A , which we shall refer to as (5.1')-(5.5'), imply that A 6 ^C(x) .
Using standard considerations we can deduce from (5.1'), (5.2') and (5.3') that the
marginal /j of A over Y is a product measure supported on A(x) , that the marginal of
/x over Y* is f*(-jx) , and that A has an r.c.p.d. v over Z such that £(-|y) is
supported on C(x,y) for all y 6 A(x) . Now consider the game with continuation paths
specified by v . The marginals \i- of \l over the Y. need not constitute a Nash
equilibrium for this game. However, by (5.4'), the set Y- of a- 6 A.(x) such that
deviation by active player i from /x. to 6 , the Dirac measure concentrated at a- , will
1 a- 1
i
raise his payoff above x. has //.—measure zero. We may therefore alter v in such a way
that it remains an r.c.p.d. of A, but such deviations are no longer attractive. We set
K- 17) = K- ly) a y. 6 Y\Y for all i , and K-|y) = *q.(X|y) if 7j e Yj\Yj for all j
^ i but y. 6 Y- . It is of no importance how we define v{ • | y) if y. e Y. for more than
one i. (Such a y corresponds to a coordinated deviation by two or more players.) For
definiteness, we set ^(-|y) equal to £(-|y) in this case. Inequality (5.5') ensures that,
with the new continuation paths specified by v , deviation is no longer profitable. The //•
therefore constitute a Nash equilibrium with these continuations, and therefore A
€ Jc(x) .
Having proved that $C has closed graph, all that remains is to show that \?C is
non-empty valued. To this end, fix x and find, for each i , a sequence {y ■} that is
dense in A.(x) . For each 1 < N < od , construct a finite game with action sets {y - 1 1 < n
< N} in the natural way from the existing game x . And for N = a> simply take the
existing game x . The family of games obtained as N varies is continuous, and its
equilibrium correspondence therefore has compact values and a closed graph. Moreover
standard results show that the set of equilibria is non-empty when N < od . It follows that
22
it is non-empty for N = cd too. □
Lemma 4.1 captures the essence of the backwards program: we begin with an upper
semicontinuous correspondence C depending on the sequence of outcomes up to and
including the current period, and end up with an upper semicontinuous correspondence 'I'C
depending on the sequence of outcomes preceding the current period. There is however one
further complication. When we come to complete the proof of the theorem in Section 4.4,
the continuation paths in Z will themselves be probability measures over sequences of
future outcomes. In the present, more abstract, setting this can be captured by writing Z
= ^^(W) , where W is itself a complete separable metric space. With this notation, the
equilibrium paths described by *C lie in 9JC{Y x $>JC(W)) . This means that #C
cannot serve directly as input for the next iteration of the backwards program. What is
required instead is a correspondence ^C : X -> 9JC{Y x W) . (The point is that elements
of &JC{Y x W) are probability measures over sequences of future outcomes, whereas
those of 9>Jl[X x 3>JC{yi)) are not.) Such a correspondence can be obtained by replacing
elements of PJC^&Jt^)) with their expectations.
More formally, for each xGX let ^C(x) c ,?1(Y x W) be the set of measures A
such that:
(i) the marginal \i of A over Y is a product measure with supp(/z) C A(x) ;
(ii) the marginal \l* of \i over Y* is fg(x) ;
(iii) A possesses an r.c. p. d. v over W such that v{ • | y) e co[C(x,y)] for all y
e A(x) ;
(iv) moreover the marginals /j. of \i over the Y. constitute a Nash equilibrium when
the continuation path following outcome y is r/(- |y) for all y .
(Once again it may be helpful to expand on (iv). Suppose that we are given \i-
6 &JC(k.{xj) and a transition probability v such that u(- |y) G co[C(x,y)] for all y
€ A(x) . Then active player i 's payoff is /u-(x,y,^(- |y))d/i(y) , where /z is the product
of the (jl- . The //• constitute a Nash equilibrium if no such player can improve his payoff
23
by changing his p. .)
Lemma 4.2 Suppose that the domain of definition of u-(x,y,z) is convex in z , and
suppose that u(x,y,z) is linear in z on its domain. Then *C is an upper
semi continuous mapping from X into Jtp(?JC(Y x W)) .
It should be noted that, when we come to apply Lemma 4.2, u-(x,y,z) will be
defined as the expectation with respect to z of an underlying function of (x,y,w) . So the
convexity of the domain of u.(x,y,z) in z , and the linearity of u-(x,y,z) in z , will be
natural consequences of the problem structure. Lemma 4.2 will be proved by showing,
roughly speaking, that #C is the image of tfC under the projection that consists of
replacing elements of &J£($>J((W)) by their expectations, and that this projection is
continuous.
Proof Let A G ^JC{Y x Z) , let \i be the marginal of A over Y , and let v be an
r.c.p.d of fi over Z. Let fj, = fi , and let u{- |y) = /z d^(z|y) be the expectation of
j/(-|y) for all y£Y. Then the projection X of A onto M(Y « W) is the probability
measure with marginal \i and conditional v . (It should be clear that X is independent
of the r.c.p.d. chosen for A .) The first step of the proof is to show that projection is
continuous.
To this end, let {An} C 9Jt{X x Z) converge to A 6 9JC{Y x Z) . Then, to show
that {A~n} C ?JC(Y x W) converges to Xe^(YxW), it suffices to show that
/x(yMw) dA~n(y,w) -. /x(yMw) dl(y,w)
for all continuous functions x- Y "♦ K and ifr. W -» IR . Let t 3>JC(W) -» K be the
continuous linear functional defined by 1(k) = /^w)d«(w) . Then
24
/x(yMw)d!n(y,w) = / Mw)di/n(w|y) x(y)d**n(y)
(by Fubini's theorem)
= /
K\(-\v)) x(y)d/*n(y)
(by definition of €}
= /
/<z)di/n(z|y) x(y)d/xn(y)
(by definition of the expectation of u (• |y))
/x(y)^(z)dAn(y)Z)
(by Fubini again, and because // = // ). But the latter expression converges to
/x(y)4z)dA(y,z) by definition of weak convergence, and this integral is equal in turn to
/x(y)V(w)dA"(y,z) by reversing the above argument. This completes the proof of
continuity.
To complete the proof, it suffices to show that gr(^C) is the image of gr(^C)
under the projection that takes (x,A) into (x,X) . Suppose first that (x,A) G gr(^C) .
Then the payoff from A is
/Uj(x,y,z)dA(y,z) = / /ui(x,y>z)di<z|y) d/x(y)
= /ui(x,y,/zd^(z|y))d/i(y)
(by the linearity of u-)
25
= /ui(x,y,K-|y))d/i(y)
(by definition of V). But u{- |y) G co[C(x,y)] because supp[*/(- |y)] C C(x,y) , and
/u-(x,y,P(- |y))d/z(y) is by definition the payoff from A~ . So it is easy to see that (x,A~)
e gr(*C) .
Suppose now that (x,X) G gr(#C) . We need to find (x,A) G gr(#C) of which
(x,A~) is the projection. To this end, let C(x,y,z) be the set of k G 9Ji{V) with
expectation z and such that supp[/t] C C(x,y) . Then C has a closed graph, and
standard considerations show that C(x,y) is non-empty iff z G co[C(x,y)] . Let
c: gr(co[C]) -» &JC(7j) be a measurable selection from the restriction of C to gr(co[C]) ,
let u{ ■ | y) = c(x,y,F( ■ | y)) for all y G A(x) , and let v{ • | y) be arbitrary for y £ A(x)
(for example, one could set v{- |y) = &-, , x for such y). Then it can be checked as
above, using the linearity of u- , that the measure A obtained by combining the marginal
Jl with the transition probability v lies in ^C(x) . □
Lemma 4.2 is the basic result justifying the backwards program. We conclude the
present subsection by relating it to the work of Simon and Zame (1990).
The standard existence theorem for normal— form games can be expressed as
follows. If each player's strategy set is a compact metric space, and if there is a continuous
function associating a vector of payoffs with each vector of strategies, then there exists a
vector of (possibly mixed) strategies that forms a Nash equilibrium. Simon and Zame
(1990) extended this result by showing that if, instead, there is an upper semi continuous
correspondence associating a convex set of payoffs with each vector of strategies, then there
exists a selection from this correspondence and a vector of (possibly mixed) strategies that
forms a Nash equilibrium when the payoff function is given by this selection.
In order to explain how Lemma 4.2 extends the result of Simon and Zame, it is
helpful to reinterpret their work slightly. Much as we do in the proof of Lemma 4.1, they
consider a sequence of finite approximations to their basic game. Suppose these
26
approximations are indexed by N , suppose that the mixed strategies /a^- constitute a
Nash equilibrium of game N for some selection from its payoff correspondence, and let the
equilibrium payoffs of this equilibrium be 7iYr- . Then their proof shows that, if the mixed
strategies /a- and payoffs ir- constitute a limit point of the strategies /a^- and payoffs
7iVr- , then there exists a selection from the payoff correspondence of the original game such
that the \i- constitute a Nash equilibrium with equilibrium payoffs ir- when the payoff
function is given by this selection. In other words, they demonstrate a limited form of
upper semicontinuity for a particular continuous family of games.
Lemma 4.2 therefore extends the result of Simon and Zame in two ways.12 The
first is relatively minor: it shows that upper semicontinuity holds for a general continuous
family of games. The second is more substantive. The result of Simon and Zame tells us
about the behaviour of the equilibrium strategies and equilibrium payoffs, but it tells us
very little about the selections from the payoff correspondence that enforce these strategies.
Lemma 4.2, by contrast, concerns a comprehensive description of equilibrium. Such a
description of equilibrium is not without intrinsic interest. But its main significance for
the purposes of the present paper is that it allows us to avoid a measurable selection
problem that would otherwise arise when we come to program forwards.13
12 As it stands Lemma 4.2 is not a direct generalisation of the result of Simon and
Zame. But it can easily be adapted to obtain one. To do so, view Z , not as a space of
probability measures over a compact metric space, but more generally as a compact
metrisable subset of a locally convex topological vector space. (This perspective is more
general, and includes the possibility that Z is a compact subset of IR . If Z is a compact
subset of IR , then one can define u-(x,y,z) = z. .) And define ^C(x) in a more limited
way, to consist of pairs (/a,7t) G &J((Y) * IR , where /a is the distribution of a Nash
equilibrium and ir is its equilibrium payoff vector. (This more limited definition of $C is
the price one must pay for the more general Z .) Then a proof almost identical to that of
Lemma 4.2 shows that $C is a continuous projection of $C, and therefore upper
semi continuous.
13 It would be tempting to conduct the backwards program as follows. Let II: gr(A)
-» IR be an upper— hemicontinuous convex— valued correspondence describing possible
continuation payoffs. Then $11: X -» IR , the correspondence describing possible Nash
equilibrium payoffs for the current period, is upper hemicontinuous; and so therefore is
co^II] , the continuation— payoff correspondence for the next iteration of the backwards
program. This simple program is, however, difficult to reverse. For, in order to reverse it,
27
4.3 The Forwards Program
In this subsection we develop the analysis necessary for the forwards program. For
this purpose we shall need a measurable selection c from #C . This function can be
thought of as part of the output of previous iterations of the forwards program. It
describes the equilibrium paths that must be followed from the current period onwards if
the strategies calculated for earlier periods by the previous iterations of the forwards
program are really to constitute part of an equilibrium.
Lemma 4.3 There exist measurable functions A: X x [0,1] -» ^J({Y * W) and w.X * Y
x [0,1] -» $>JC{W) such that, for all (x,£) e X * [0,1] :
(i) the marginal /x(-|x,£) of A(-|x,£) over Y is a product measure such that
supp[/x(-|x,0] C A(x) ;
(ii) themarginal^0(-|x,£)of/z(-|x,£) over Y0 is ity{-\x);
(iii) i/(-|x,-,£) is an r.c.p.d. of A(«|x,£) over W such that u{- |x,y,£) € co[C(x,y)]
for all y 6 A(x);
(iv) the marginals /j.(-|x,£) of /x(»|x,£) over the Y- constitute a Nash equilibrium
when the continuation path following outcome y is v{- |x,y,£) for all y e A(x).
Moreover /A(- |x,£)d£ = c(x) for all x e X .
Lemma 4.3 captures the essence of the forwards program. It requires some
explanation. First of all, conditions (i)— (iv) imply that A(- |x,£) € ^C(x) . The condition
/A(- |x,£)d£ = c(x) therefore tells us that, if the equilibrium path is chosen from ^C(x)
on the basis of the current public signal £ e [0,1] , then the overall path will be precisely
one must be able to find, for any measurable selection ir from #11 , a measurable
selection $7r from II such that the game with payoff function $7r(x,-) has a Nash
equilibrium with payoff 7r(x) . This is not a standard measurable— selection problem. For
what is actually required is that we select, for each x , a function $7t(x,-) , and that these
choices of functions be coordinated in x to obtain a single measurable function 3>7r . (A
version of this non— standard problem did arise for Mertens and Parthasarathy (1987;
Lemma 1 of Section 6, pp 41— 42). In their case, however, the functions $7r(x,-j could be
taken to be continuous.)
28
c(x) . (If Lebesgue measure is the marginal distribution over [0,1] , and if A(- |x,-) is the
conditional distribution over Y x W , then /A(-|x,^)d^ is the marginal distribution over
Y x W .) Secondly, the lemma goes beyond the simple assertion that there exists A: X
x [0,1] -» ?JC{Y x W) such that A(- |x,£) G *C(x) for all (x,£) . For, while this
assertion would automatically imply the existence of a v that was measurable in y for
each (x,£) , the proposition delivers a v that is measurable in (x,y,£) jointly. Thirdly,
the \i-\ X x [0,1] -• &J((Y.) are precisely the strategies of the players for the current
period. Fourthly, the restriction $c of v to gr(A) x [0,1] describes the equilibrium
paths that must be followed from the next period onwards. It serves as the input for the
next iteration of the forwards program.
Proof Most of the considerations necessary for the proof have already arisen in the proof,
of Lemmas 4.1 and 4.2. The present proof will therefore be brief.
The first step is to find a measurable d: X -• 9JL ( $>JC(Y * W)) such that, for all
x G X , supp(d(x)) c \PC(x) and the expectation of d(x) is c(x) . This can be done using
an argument very similar to one used in the proof of Lemma 4.2. Next, we need a
measurable mapping A: X x [0,1] -» &J£(Y x W) such that, when [0,1] is given Lebesgue
measure, the distribution of the random variable A(- |x,«): [0,1] -» 9Jl{Y x YV) over
^Ji(Y x W) is precisely d(x) for all x . Such a mapping can be obtained by applying a
'measurable' representation theorem, e.g. Gihman and Skorohod (1979; Lemma 1.2, p 9).
The third step is to find a measurable mapping u: X x Y x [0,1] -» &Jt(W) such that
£(-|x,-,£) is an r.c.p.d. of A(- |x,£) for all (x,£) e X x [0,1] . In other words, we need a
'measurable' version of the standard result which asserts the existence, for each (x,£) , of
an r.c.p.d. v{- |x,-,£) of A(- |x,£) . Since this is, in effect, the central step of the
forwards program, and since we have not been able to find precisely the result we need in
the literature, we sketch a proof of this.
Let /x(- |x,£) be the marginal of A(« |x,£) over Y . Let {B 1 1 < n < <d} be a
sequence of sets that generates the Borel a— algebra on Y . For each N , let 5?™ be the
29
a-algebra generated by {B 1 1 < n < N} . And let v be a fixed element of ^l#(W) .
Since &■** is finite we can give an explicit formula for an r.c.p.d. of A(-|x,£)
given the a— algebra 5?-^ on Y . For each y G Y and each measurable W C W we set
,,;, - A(YxW|x,Q
"N(W|x,y,£) = -L- - — l-x^L
M(Y|x,0
if Y is the atom of &N containing y and /i(Y|x,£) > 0 , and
N
z,N(W|x,y,£) = KW)
otherwise. It is immediate from these formulae that u-^(- |x,y,£) is jointly measurable in
(x,y,£) , and therefore that the convergence set K of u-^ is measurable. Set v
= lim!/v,on K and v — v outside K .
Certainly u is measurable. Also, since i/^(-|x,-,£) is effectively a version of the
conditional expectation of A(- |x,£) given &„ , the martingale convergence theorem
implies that 1 im i/M(- |x,-,^) exists //(• |x,£) — a.s., and that u(-\x,',£) is a version of
what is effectively the conditional expectation of A(- |x,£) given the Borel a— algebra on
Y. That is, u{- |x, •,£) is an r.c.p.d. of A(-|x,£). This completes the third step.
There are two difficulties with u as it stands: it may not be the case that
u(- |x,-,£) e co[C(x,y)] for all y 6 A(x) ; and the marginals /z(-|x,£) of A(-|x,£) need
not constitute a Nash equilibrium when the continuation path following outcome y is
u(- |x,y,£) . The fourth and final step constructs a v that does have these properties from
v . To obtain the first property, let q be any measurable selection from co[C] , and
redefine i>(-|x,y,£) to be q(x,y) wherever it does not lie in co[C(x,y)] for some y
G A(x) . To obtain the second: for each i , let q. be the function defined in the proof of
Lemma 4.1; let 7r.(x,£) = Ju-(x,y,v(- |x,y,^))d/i(y|x,^) ; and let Y.(x,£) be the set of a-
30
6 A-(x) such that /u.(x,y\a.,^(- |x,y,£))d/z(y|x,£) > ^(x,^) . Then we may set
v{- |x,y;0 = qi(x,y) if 7. 6 Y^YjCx.O for all j * i but yj £ Y(x,0 , and v{- |x,y,£)
= £(• |x,y,£j otherwise. Standard considerations show that the resulting v is measurable,
and the fact that A(- |x,£) e C(x) implies that u(- |x,»,£) is still an r.c.p.d. of
A(-|x,£). o
4.4 Completion of the proof
Before proceeding, we need to define the equilibrium correspondence for a family of
games. In making this definition we encounter a minor technicality. We have defined an
equilibrium as a strategy profile that is in Nash equilibrium in every subgame. But a
strategy profile specifies players' behaviour for the entire family of games that they might
face. Hence an equilibrium will generate a whole family of paths, one for each game x
6 X . We therefore define E.. (x ) , the set of equilibrium paths of game x , to be the set
of paths A such that A is the equilibrium path generated in game x by some
equilibrium. Similarly, reinterpreting our basic family of games as a family parameterised
t— 1 t— 1
by x e X , we can define, correspondences E. for t > 2 .
Theorem 4.1 Suppose that (Al)— (A3) hold. Then E, is an upper semi continuous map
from Xt_1 into J6{9Jt{**_ . Yj).
The main interest of Theorem 4.1 derives from the fact that it implies existence
(see Section 5). However upper semicontinuity is itself of interest: it is reassuring to know
that our concept of equilibrium possesses this standard regularity property.
Proof The proof proceeds in three steps. Let B.(x — ) be the set of paths beginning in
period t that are consistent with Nature's strategy. The first part of the proof shows that
B. is an upper semicontinuous map from X — into JGi^JC^ ®Y )). In the second, a
I S — X s
sequence of correspondences {C. 1 1 < t < cd} is constructed, each of which is upper
31
semicontinuous. In the third it is shown that E. = co[C.] for all t .
For notational convenience, we show that B1 is upper semicontinuous. To this
end, let B?(x ) be the set of marginals over « ,Y of measures in B,(x ). Then a
backwards programming argument similar to, but simpler than, that developed in Lemmas
4.1 and 4.2 shows that B, is an upper semicontinuous map from X into
J6{9Jt{* _, Y )). But a sequence of measures {A } C ^("/iY.) converges iff its
marginals over PJCi* _.Y ) converge for all t. So B, , too, is upper semicontinuous.
This completes the first step.
T
Turning to the second step, let T > 1 be given. For each t > T , let C. = B. .
T T
And, for each 1 < t < T , let C . be defined inductively by the formula C , =
T T t— 1
$, (C . , , ) . (In other words, C . (x ) is the set of equilibrium paths that are obtained
T
for period t onwards when the continuation paths can be chosen from co[C, ,] .) Then
T
the backwards program ensures that each C, is upper semicontinuous.
T T S
Now define C. = nt™,C. for each t > 1. It is obvious that C, cC. whenever T
> S. Hence C, too is upper semicontinuous. It is also the case that ^t(C, , ,) C C.. To
see this, let A 6 [^t(C. , 1)](x ) . By definition, A can be enforced by continuation
T
paths from C, , . But paths in C. ,, belong a fortiori to C, ,-, . Hence
A 6 [^t(C^+1)](xt_1) for all T . Since Ct = nT=i*t(C T+l^ ' the rec*uired conclusion
T
follows. Finally, C. C *+(C. , ,) . To see this note that the correspondences C . , , for
1 < T < od together with C,,, for T = cd form a continuous family of games with
parameters x and T . And Lemma 4.2 shows that the equilibrium correspondence of
such a family is upper semicontinuous. It follows that any A 6 C. (x — ) , which is
certainly a limit point of points in [^.(C. ,)](x ), also lies in [*t(C, , ,)](x ).
Overall, then, we obtain a sequence of upper semicontinuous correspondences
{C.| 1 < t < a) } such that C, = ^t(C. , ,) for all t . It remains only to show that E,
= co[C,] for all t . We do this for t = 1 only. The other cases are analogous.
We show first that co[CJ C E, . It suffices to show that any measurable selection
c, from co[CJ is also a selection from E1 . So let C. be any such selection. Then
32
repeated application of the forwards program starting with c, yields, for all t > 1 ,
strategies f,. and a measurable selection c.,, from co[C.,J such that: (i) for all
(x ,()e H~ , the f.-(x — ,£ ) constitute a Nash equilibrium in period t when the
continuation paths are given by c, , ,(x ,•>£ ) and (") the paths obtained from period t
onward when the f.. are employed in period t and the continuation paths are given by
c.,, are precisely those required by c. . It follows from (i) and a standard dynamic
programming argument that the strategy profile f constitutes an equilibrium. And it
follows from (ii) that the equilibrium path in game x is precisely c, (x ) . This
completes the proof that co[CJ c E,.
We now show that E, C co[CJ . To this end, let f = <f.. |t > 1, i € X<> be
any subgame-perfect equilibrium in extended strategies. For each t > 1 , let c ,: H ~~
-» 3>JC(X * Y ) describe the paths followed from period t onwards when the strategy
profile f is employed. The mapping c, inherits measurability from f . Let T > 1 be
T
given. Now certainly c™,, is a selection from co[Cy,,]. Also, suppose that c,,1 is a
T
selection from co[C. J for some 1 < t < T. Since f is an equilibrium, the f,- specify
Nash equilibria for period t when the continuation paths are given by c. , . It follows
T
at once that c, is a selection from co[C , ] . Proceeding inductively we therefore conclude
T
that c, is a selection from co[C, ] . Moreover, allowing T to vary, we obtain that C, is
a selection from co[CJ . This completes the proof. □
33
5 Existence with and without Public Signals
This section provides a systematic discussion of the question of existence of
equilibrium in the dynamic model introduced in Section 3. It begins with a recapitulation
of the main result for this case, namely that, when public signals are allowed, any family of
games satisfying the standing assumptions (Al)— (A3) possesses an equilibrium. However,
the section also addresses two further questions. Under what circumstances can existence
be obtained without the use of public signals? And is the extension of the concept of
subgame— perfect equilibrium obtained by allowing the observation of public signals
sufficiently rich?
5.1 Existence with Public Signals
The following result is an immediate corollary of Theorem 4.1:
Theorem 5.1 Suppose that (Al)— (A3) hold. Then there exists a subgame-perfect
equilibrium in extended strategies.
Proof Pick any x eX . By Theorem 4.1, E,(x ) is non-empty. Pick any A 6 E.(x ).
Then, by definition of E..(x ) , there exists an equilibrium of the family of games that
generates equilibrium path A in game x . In particular, there exists an equilibrium!
The potential drawback of this result is that, in order to obtain the existence
equilibrium in what is intended to be a purely non-cooperative situation, it introduces an
extraneous element into the model, namely public signals. It is therefore of some interest
to discover circumstances under which equilibrium exists even without this element.
34
5.2 Existence without public signals
There are at least three classes of game in which public signals are not needed for
existence: games of perfect information, finite— action games, and zero— sum games. The
results for such games are summarised in Theorems 5.2, 5.3, and 5.4 respectively.
Theorem 5.2 Suppose that (Al)— (A3) hold, and that the family of games has perfect
information. Then a subgame— perfect equilibrium exists. a
Proof Oddly enough, Theorem 5.2 is not an immediate corollary of Theorem 5.1.
(Intuitively it would seem obvious that, when only one player acts at a time, public signals
do not allow players to achieve anything they could not achieve using mixed strategies.
But there is a minor complication: players observe not only the current signal, but also
past signals.) However, an equilibrium can be constructed using the correspondences C,
used in the proof of Theorem 4.1. We begin with any measurable selection c, from C-. ,
and define f,.(x ) to be the marginal of c,(x ) over Y... . Next, let c« be a measurable
selection from co[C J that enforces c, . Since the game is a game of perfect information,
co[C2] = C^ • Hence c„ is also a measurable selection from C« . So we may define
L.(x ) to be the marginal of c2(x ) over Y,. . And so on. o
Indeed, one can even show that there exists a subgame-perfect equilibrium in
which players employ pure actions only. We do not demonstrate this here, since the
argument does not appear to be a simple corollary of the construction that we have
employed.14 Note that the proof of Theorem 5.2 shows incidentally that allowing public
signals in a game of perfect information does not enlarge the set of equilibrium paths.
14 See Harris (1985) for a treatment of the case of perfect information. The argument
used there can easily be adapted to the present context. Alternatively, one can work
directly with the apparatus used here, taking care to purify the selections as one programs
forward.
35
Let us say that a family of games is of finite action if X is finite, and if, for all
t > 1 and all x" 1 6 X1 * , A^x1 *) is a finite set.
Theorem 5.3 Suppose that (Al)— (A3) hold, and that the family of games is of finite
action. Then a subgame— perfect equilibrium exists. Moreover, for each x 6 X , the set
of equilibrium paths is compact.
Proof Theorem 5.3 is more or less well known, and there is more than one way of proving
it. (See Fudenberg and Levine (1983) for a different proof.) So we merely note that it can
be proved by a simple adaptation of the methods of this paper. The basic idea can be
expressed in terms of the notation of Section 4.2 as follows. Instead of defining #C(x) to
be the set of equilibrium paths that can be obtained by choosing continuation paths from
co[C] , one defines it to be the set of equilibrium paths that can be obtained by choosing
continuation paths from C. Because X is finite, and because the A.(x) are finite for all
x e X , this new definition still yields an upper semicontinuous correspondence.
It should be noted that both finiteness assumptions are needed. Indeed, suppose
that X is finite but that one or more of the A-(x) are infinite for some x. Then it can
happen that there is a sequence of equilibria of game x, each of which involves a public
signal generated endogenously by players' randomisation over their action sets A.(x), and
that this sequence converges to a limiting equilibrium which does not generate any such
signal. Suppose, on the other hand, that the A.(x) are finite for all xgX but that X is
infinite. Then it can happen that there is a convergent sequence of games in X , and that
there is at least one player for whom at least one player for whom at least two actions
coalesce in the limit. If this player randomises over his two actions, then he generates a
public signal which disappears in the limit.
It is precisely in order to compensate for the potential disappearance of such
endogenously generated public signals that the exogenous public signals are used in our
basic existence theorem. o
36
To complete our discussion of the existence of subgame— perfect equilibrium, we
consider the case of a family of zero— sum games. (A family of games is of zero sum if J<A
contains precisely two elements, and if E. j/^\ = 0 .)
Theorem 5.4 Suppose that (Al)— (A3) hold, and that the family of games is of zero sum.
Then a subgame— perfect equilibrium exists.
In order to understand this result, it is helpful to recall some facts about one-
period zero— sum games. It is well known that each players' set of optimal strategies is
convex in such a game. But the set of equilibria, regarded as the set of probability
distributions over outcomes, need not be. Hence, even in a zero— sum game, the set of
equilibria will be enlarged if players are allowed to observe a public signal. The set of
equilibrium payoffs, on the other hand, will not. For the public signal merely allows the
two players to co-ordinate on a choice of Nash equilibrium for the game, and all Nash
equilibria have the same equilibrium payoff (namely (v, — v) , where v is the value of the
game).
Proof Note first that our proof of the existence of equilibrium shows, in particular, that
the extended family of games has a Nash equilibrium. It follows immediately from
standard considerations that this family also has a value. That is, there exists a
measurable mapping v, : X -> [R such that v, (x ) is the value of extended game x .
(Indeed, since v.,(x ) is simply player l's payoff from any path in E,(x ), and since E, is
upper semicontinuous, v, is actually continuous. We do not need this fact, however.)
Similarly, there exist mappings v ■ X ~~ -» IR such that v.(x ) is the value of the
extended game beginning in period t.
With these considerations in mind, we can construct a subgame— perfect
equilibrium of the original family of games as follows. The construction follows the
construction of an equilibrium of the extended game, but takes care to avoid the need for a
37
public signal. The first step is to find a measurable selection c, from C, . Since c, is a
measurable selection from C, rather than co[C,] , its marginals over Y, are already
product measures. So we may define strategies f,.: X -» ^^(Y.-) by setting f,.(x ) to
be the marginal of c,(x ) over Y... . Similarly, c, generates a measurable selection Cr,
from co[CJ that enforces the f,. . But all A e co[C2(x )] generate the same
continuation payoff, namely v2(x ) . In particular, if c« is a measurable selection from
C9 then (L generates the same continuation payoffs as c« • Hence the f, • can be
enforced by <L just as well as by c« • (The continuation paths specified by c« do not
necessarily constitute an r.c.p.d. of c, , of course.) Iterating this argument we obtain
strategies £.: Xt_1 -» ?JC{Y.^ for all t > 2 as well, o
5.3 Approximation by finite— action games
Suppose that we are given a single game (i.e. a family of games for which X is a
singleton) satisfying (Al)— (A3). Then a finite— action approximation to this game is,
roughly speaking, a new family of games indexed by n e IN U{od} such that: the game
corresponding to n = oo is the original game; for each n 6 IN , the game corresponding to n
is of finite action; and taken as a whole the new family of games satisfies (Al)-( A3). (Here
(Al) ensures that players' action sets in the original game are continuously approximated;
(A2) ensures that Nature's behaviour is continuously approximated; and (A3) ensures that
active players' payoffs are continuously approximated.)
Finite-action approximations to a single game satisfying (Al)-(A3) certainly exist.
Indeed, we can even construct a finite-action approximation to a family of games satisfying
(Al)-(A3). To do this, we need the following ingredients. First, for each t > 1 and each
i 6 J, let {yjj|n 6 IN} be dense in Yt- ; and let {yj|n 6 IN} be dense in YQ = X°.
Secondly, for each n € IN , let Y|| = {ym| 1 < m < n} . Thirdly, for each t > 1, i G J and
n 6 IN, let o£j:X — -» Y.. be a continuous function such that, for all x — e X — :
a..(x — ) belongs to A,.(x — ) , and ay(x — ) is as close to y?. as any other point in
A..(x ). (Such a function exists because A,.:X — -» J£(Y..) is continuous. In
38
n vt-l
Nature's case the function is simply constant with value y.* .) And let a.-: X
-. MYti) be defined by ^(x1-1) = {aj^x*-1)! 1 < m < n} for all xt_1 e Xt_1.
Fourthly, for each t > 1 and n 6 IN, let ^\*: X — -» &JC(Y.q) be a continuous function
such that, for all x e X : supp [</>t(j(x )] C {y . A\ < m < n}; and <Ptrt(x ) is as
close to ffrt(x ) as another measure in &J((Y. *) whose support is contained in
{y™0 1 1 < m < n} . Finally: let Yj = XQ ; for all t > 1 , i 6 Jj6 and xt_1 6 Xt_1 , let
^.(x1-1) = Ati(xt_1) ; and for all t > 1 and xt_1 e Xt_1 let ^0(xt_1) = ft0(xt_1) .
In terms of these ingredients we can define a larger family of games, which can be
interpreted as a sequence of families of finite-action games approximating the original
ssr\
family, as follows. First, let X be the union over n 6 IN U {qd} of the sets {n} * Yn .
^t— 1 ^
Secondly, suppose that X has been defined for some t > 1 . Then: for all ie 7, A,.
S\i 1 s\ s\ /\ /\
is the restriction of aj-(-) to X — ; A. = x- A • and X. = gr(A.) . Thirdly, for all
t > 1 , f.0 is the restriction of fl^-) to X — . Finally, payoff functions u- are defined
in the obvious way.
Now suppose that we are given a finite-action approximation to a single game.
Since this approximation is simply a family of games parameterised by n e IN U {qd} and
satisfying (Al)-(A3), its equilibrium correspondence is upper semicontinuous in n. Hence,
since any subgame-perfect equilibrium path of any game n 6 IN is, in particular, an
equilibrium path of that game, it follows that any limit point of subgame-perfect
equilibrium paths of the finite-action games is an equilibrium path of the original game.
And our solution concept (namely subgame-perfect equilibrium in extended strategies)
includes everything that can be obtained by taking finite-action approximations to a game
(or, more precisely, everything that can be obtained by taking continuous finite-action
approximations to a game satisfying (Al)-(A3)).
5.4 Approximation by e-equilibria
One can also show that any limit point of subgame-perfect e-equilibrium paths of
a given single game is an equilibrium path of the game, by combining the techniques of this
39
paper with those of Borgers (1989). Indeed, suppose that we are given a family of games
satisfying (A1)-(A3). For each t > 1, xt_1 e Xt_1 and e > 0 , let Et(e,xt_1) be the set
of equilibrium paths of subgame x of a family of games identical to the original game
except that: (i) each player i G Jji is replaced by a sequence {t-|t > 1} of agents; (ii)
each agent chooses actions that are at or within e of being optimal in each subgame.
Then it is easy to adapt the methods of Section 4 to show that E. is upper
semicontinuous. But this implies the required result. For any subgame-perfect
e— equilibrium path of the original game is also an e-equilibrium path; any e— equilibrium
path is an e— equilibrium path of the game in which each active player is replaced by a
sequence of agents; and the 0— equilibrium paths of the game in which each active player is
replaced by a sequence of agents coincide with the equilibrium paths of the original game.
40
7 Conclusion
The results of this paper concerning the existence of subgame-perfect equilibrium in
general dynamic games create a small dilemma. If one adopts the point of view that the
observation of public signals violates the spirit of non-cooperative game theory, then the
interpretation of the counterexample of Section 2 is presumably that games with a
continuum of actions are somehow intrinsically pathological. After all, that example seems
wholly non— pathological in all other respects. If, on the other hand, one thinks that games
with a continuum of actions are an indispensable part of non-cooperative game theory,
then presumably one must accept that games must be extended to allow the observation of
public signals.
Faced with this dilemma it is tempting to dismiss games with a continuum of
actions as pathological. There are, however, at least two reasons why this reaction may be
too simplistic. First, it can be argued that a solution concept should be robust to small
errors in the data of the games to which it applies, in the sense that one should not rule out
equilibria that occur in games arbitrarily close to the game under consideration.15 (In other
words, the equilibrium correspondence should be upper semicontinous.) Provided that such
errors are confined to payoffs, the solution concept of subgame-perfect equilibrium satisfies
this requirement when applied to finite— action games. If, however, such errors may be
present in the extensive form itself, then public signals must be introduced if upper
semicontinuity is to be satisfied. Secondly, it is not entirely clear that public signals really
do violate the spirit of non-cooperative game theory. After all, players must in any case
coordinate on a specific Nash equilibrium to play; and public signals merely increase the
extent to which they can coordinate, by allowing them to coordinate on a random Nash
equilibrium.
15 See Fudenberg, Kreps and Levine (1988) for example.
41
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